This is how we are taught.
In training you should grip tight enough so that nage struggles to do the technique, then you should let off a bit so they can do it. Over time the technique improves bit by bit and the grip can get stronger and stronger. This is the Iwama Ryu way of doing things.
Uke gets stronger, nage's technique improves.
You did the right thing to not deviate to a different technique. The uke should, in my opinion, let off the tension a bit and let you do the technique to the level you are capable of. This is how you improve.
In future, maybe ask them to hold a little easier? So you can actually do the technique.
What is that grab supposed to be? If the intention is to train a good martial artist, then teaching people to do that ridiculous "grab of death" is silly. The first thing any martial artist from another style will say when he sees Aikido is that no one attacks that way. And it is true, no one outside of Aikido attacks that way.
Who ever won a fight by keeping the other guy from moving? You win a fight by breaking his balance, putting him in a position in which he cannot defend himself, and striking him. Not only does grabbing hard focus all the power on the wrist rather than the center, the tension involved curtails the freedom of movement of the attacker as well as the defender. This is not intelligent martial arts. Try taking someone's balance using strength. I am a 250 pound guy. If I grab someone my own size there is no way I can take his center if I tense up my arms when I grab.
On the other hand, I can have a person off balance and struck two or three times if I keep my arms relaxed and don't think this has anything to do with stopping someone else's movement. Since Aikido is being presented as some sort of extension of pub crawling or defense against foot ball hooligans, I'll say that in any kind of "applied" situation, I can't think of a less applicable skill than grabbing someone so hard their hand turns purple. Absolutely no function.
With my students, we start with katatetori and have the nage throw a punch with the off-hand. Or a kick... The uke should be able to use the grabbing hand to "solve" that problem. Try doing that while squeezing hard. Then we teach the uke to grab, use the grab to break nage's balance, and strike him. The instant you grab some the way you are talking about, you are totally open and cannot defend.
It is simply a fact that tension slows you down, reduces your power, and restricts your freedom. This is one of the reasons that serious folks from other martial arts think Aikido is bullshit. They look at this grab the wrist stuff and see what a joke it is. You'll hear that these grabs were originally about stopping someone from accessing his weapon, usually sword. But actually, it wasn't about stopping him from pulling his sword, it was about using the grab to break his balance and take the sword away from him and cutting him with it. That is entirely a different matter.
Now there are folks who have done a lot of IP work that can grab you in a completely relaxed manner and you'll have a hard time moving. That's different as they are using their structure for that strength and not doing anything which restricts their freedom of movement or makes it hard to defend themselves. But this entails an understanding of how to direct power to the nage's center and has nothing to do with holding the wrist hard. I can seriously restrict someone's ability to move while my arms are relaxed and my grip is only slightly tight, pretty much as you'd hold a sword. But, outside of kihon waza training in which the point is feedback for the nage, I would not see restricting the opponent's movement as having much utility at all. Do that and it's like grabbing an anchor... who actually has who? I render you unable to move using muscle strength and I am just as unable to move until I release you.
Everyone seems to basically agree that sword and empty hand are related. well, tell me when you would ever use the kind of strength and tension with sword? Pretty much never. So we shouldn't be imprinting that kind of mistaken tension through our training. You want to see what grabbing properly should look like, take a look at the YouTube clips of Judo's Mifune. That's grabbing.